Shunt Model

The shunt model treats each electrode $e_{\ell }$ as a perfect conductor. The potential is constant on the electrode, and only the total current through it is prescribed (see Electrode Models):

$$\begin{array}{rlrl}\nabla \cdot (\sigma \nabla u)& =0& & \text{in }\Omega ,\\ u& =U_{\ell }& & \text{on }{e}_{\ell },\\ {\int }_{e_{\ell }}\sigma {\partial }_{\nu }u\mathrm{d}s& =I_{\ell }& & \ell =1,\dots ,L,\\ \sigma {\partial }_{\nu }u& =0& & \text{on }\partial \Omega \setminus {\bigcup }_{\ell }{e}_{\ell }.\end{array}$$

The electrode voltages $U_{\ell }$ are unknowns, as in the Complete Electrode Model. The shunt model is the limit $z_{\ell }\to 0$ of the CEM (no contact impedance).

Edge singularity. At the boundary of an electrode, the boundary condition switches from Dirichlet to Neumann. There the current density is singular, like $r^{-1/2}$ in the distance $r$ to the electrode edge. Most of the current enters near the electrode edges, not uniformly as the Gap Model assumes. A positive contact impedance removes this singularity, which is one reason to prefer the CEM.

Voltage-driven form. Prescribing the electrode voltages $U$ instead of the currents gives a mixed boundary value problem: Dirichlet on the electrodes, homogeneous Neumann in the gaps. It is uniquely solvable without grounding, and the currents follow as $I_{\ell }={\int }_{e_{\ell }}\sigma {\partial }_{\nu }u$. Discretely, $u$ is fixed on all nodes of $e_{\ell }$, and $I_{\ell }$ is the sum of the residual of the discrete equation over these nodes (see Discrete Electrode Models).

The gap and shunt models are different models. The voltage-driven shunt problem is not the inverse of the current-driven gap model: the gap model has a uniform current density with a non-constant potential on the electrode, the shunt model the reverse. Functionals that compare a current-driven and a voltage-driven solution of the same data, such as the Kohn-Vogelius Functional, therefore do not vanish at the true conductivity if these two models are paired.

In ModularEIT.jl: GapModel, forward_dirichlet.

References

1. K.-S. Cheng, D. Isaacson, J. C. Newell, D. G. Gisser (1989). Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36(9), 918–924. doi:10.1109/10.35300
2. E. Somersalo, M. Cheney, D. Isaacson (1992). Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography. SIAM J. Appl. Math. 52(4), 1023–1040. doi:10.1137/0152060